Conversion filtering for digital signaling systems

ABSTRACT

One way to increase the bandwidth efficiency of a linear modulation scheme is to increase the number M of possible signaling levels. Such an increase is achieved however at the expense of a smaller minimum distance d min  between the multilevels, for a fixed average signal power. The present invention involves the use of a conversion filter as part of a digital signaling system used over a communications channel of limited bandwidth and power. A conventional linear digital signaling system consists of an encoder, a shaping filter, a modulator and a bandpass filter. In the present invention, the digital signaling system consists of an encoder, a conversion filter, a modulator and a passband filter. The conventional filter increase the bandwidth efficiency of the scheme when its impulse response spreads over L signaling intervals. Such an increase is achieved however, at the expenses of a smaller d min . By increasing the modulation index h of the conversion filter, it is possible to increase d min . The invention also provides steps for choosing the modulation index h, the cutoff frequency f oo  of the bandpass filter and the spread L of the conversion filter such that d.sub. min is locally maximized for a fixed bandwidth efficiency and a fixed complexity of the receiver.

BACKGROUND OF THE INVENTION

This invention relates to the processing of digital data for transmission over a communications channel of limited bandwidth and power, and relates to a method and apparatus for increasing the bandwidth efficiency and the power efficiency of the transmission.

One way to increase the bandwidth efficiency of a linear modulation scheme is to increase the number M of possible signaling levels. Such an increase is achieved however at the expenses of a smaller minimum distance d_(min) between the multilevels, for a fixed average signal power.

Csajka et al (U.S. Pat. No. 4,077,021) describe Trellis Coded Modulation (TCM) which combines convolutional coding with linear multilevel modulation schemes in an efficient way. TCM preserves the Bandwidth (BW) efficiency of the modulation schemes while offering 3 to 6 db increase in d_(min) ² over multilevel modulation, and is considered the state of the art in the domain of power and Bandwidth (BW) efficient modulation schemes.

The present invention involves the use of a conversion filter and a bandpass filter (BPF) together with an encoder and a modulator as part of a digital signaling system. The difference between the signaling system involved in the present invention and the signaling system comprising TCM is the conversion filter which allows the signaling system to generate schemes that are more BW-efficient than the schemes generated by the encoder-modulator combination. It can be shown that the digital signaling system described in the present invention can generate schemes that have larger d_(min) ² /E_(b) than TCM for comparable BW-efficiency and comparable complexity of the receiver, E_(b) being the average energy per transmitted bit.

Conversion filtering was first introduced in 1969 by Lender (Canadian Patent No. 823,307,2) in order to generate a signal of near constant envelope. In Lender, the conversion filter is an Infinite Impulse Response (IIR) filter, constrained to have a specific Power Spectral Density (PSD) and a modulation index h equal to 1/2. Subsequently, conversion filtering was used to generate signals of constant envelope (A. R. Hambley and O. Tanaka, "Generalized Serial MSK Modulation," IEEE Trans. on Commun., Vol. COM-32, pp. 305-308, March 1984 and F. Amoroso and J. A. Kivett, "Simplified MSK Signaling Technique," IEEE Trans. Commun., Vol. COM-25, pp. 433-441, April 1977). In both cases, the conversion filter is a Finite Impulse Response (FIR) filter, constrained to have a length equal to one symbol duration, a modulation index h equal to 1/2 and whose Lowpass (LP) equivalent Impulse Response (IR) g(t) is restricted to take the format ##EQU1## where φ(t) is a slowly varying function and j=√-1.

SUMMARY OF THE INVENTION

In one aspect, the present invention provides an apparatus for transmitting a digital signal over a communications channel, the apparatus comprising:

at least one encoder for converting digital signals into a sequence of symbols;

a conversion filter electrically connected to the encoder and having an impulse response with a spread L and a modulation index h, for filtering the sequence of symbols to generate a signal S;

a modulator electrically connected to the conversion filter for modulating the signal S to generate a signal V;

a bandpass filter electrically connected to the modulator and having a cut-off frequency f_(co) for filtering the signal V to generate a signal R;

the spread L, the cut-off frequency f_(co) and the modulation index h being chosen so that the power efficiency is locally maximized for a fixed bandwidth efficiency.

In another aspect, the present invention provides a process for transmitting a digital signal over a communications channel, the process comprising:

encoding the digital signals to generate a sequence of symbols;

filtering the sequence of symbols with a conversion filter to generate a signal S, the conversion filter having an impulse response with spread L and a modulation index h;

modulating the signal S with a modulator to generate a signal V;

filtering the signal V with a bandpass filter to generate a signal R, the bandpass filter having a cut-off frequency f_(co) ;

the spread L, the cut-off frequency f_(co) and the modulation index h being chosen so that the power efficiency is locally maximized for a fixed bandwidth efficiency.

A further summary of the invention is found in the claims appended to this Patent.

BRIEF DESCRIPTION OF THE DRAWINGS

There will now be described the manner of operation of the invention and preferred embodiments of the invention, with reference to FIG. 1, which is a block diagram of the 4 processing stages of the digital signaling system according according to the invention, and

FIG. 2 shows a graph of power spectral density of a conversion filter with Gaussian impulse response for utilization in accordance with the invention.

DESCRIPTION OF THE INVENTION

The present invention involves a digital signaling system which comprises an encoder, a conversion filter, a modulator and a BPF as shown in FIG. 1. The encoder receives a sequence α of bits intended for transmission and generates a sequence β of symbols. The sequence α consists of the bits . . . , α_(i), α_(i-l), . . . which take one of the two values ±1 and the sequence β consists of the symbols . . . , β_(i), β_(i-1), . . . , where i is an integer taken from { . . . , -1, 0, 1, . . . }. Possible encoders in FIG. 1 are: precoders, channel encoders and source encoders, or any combination thereof. In the present invention, precoders are used essentially to prevent error propagation, channel encoders are used essentially to achieve a gain in d_(min) ² and source encoders are used essentially to achieve a gain in the BW efficiency. Examples of precoders are differential encoders where β_(i) =(α_(i) +β_(i-1))modulo-M. Examples of channel encoders are block encoders and convolutional encoders (see J. G. Proakis, "Digital Communications," McGraw-Hill, New York, 1983).

Examples of source encoders are M-ary mappers followed by a Gray encoder where the real part of β_(i) takes one of the M values {±1, ±3, . . . ±(M-1)} (assuming M even) and where the imaginary part of β_(i) takes either the value 0 or one of the M values {±1, ±3, . . . , ±(M-1)}. When the imaginary part of β_(i) takes only the value zero, β_(i) is said to be real. When the imaginary part of β_(i) takes one of the M values {±1, ±3, . . . , ±(M-1)} β_(i) is said to be complex.

In FIG. 1, the conversion filter receives the sequence β and generates the signal S'(t) which is a function of time "t" and of the sequence β. The signal S'(t) is defined ##EQU2## where T is the signaling interval which corresponds to the duration of one symbol and g(t) is the lowpass equivalent impulse response of the conversion filter. The preferred shape for g(t) is a Gaussian function of time defined as

    g(t)=(4πσ.sup.2).sup.1/4 e.sup.-2(πσ).spsp.2 e.sup.j2πht/T

where h is the modulation index which determines the frequency by which the conversion filter is offset from dc (i.e. from the zero frequency) and σ is inversely proportional to the spread L. FIG. 2 shows power spectral density of the conversion filter for various choices of L. The spread L is defined as the number of signaling intervals over which g(t) spreads in a significant manner, i.e. in a manner where g(t) is smaller than essentially about 10⁻² for t larger than LT.

While a preferred manner of applying the conversion has been described, any alternate conversion filter having the following preferred complex autocorrelation: "e.sup.(-πσ).spsp.2 e^(j2)πht/T ", can be used to generate S'(t).

See attached software for a display of the conversion filtering process. The software consists of a computer simulation of the encoder (noprecod.f), of the conversion filter (Gaus WMF.f), of the modulator and of the bandpass filter (channel WMF.f) according to the above preferred specifications. The communications channel is an Additive White Gaussian Noise channel.

In FIG. 1, the modulator receives the signal S'(t) and generates the signal S(t) defined as

    V(t)=Re{S'(t)e.sup.j2πf.sbsp.c.sup.t }

where f_(c) is the carrier frequency assumed to be much greater than 1 /T and "Re{α}" denotes the real part of "α". In other words, S'(t) is the complex envelope of S(t).

Finally, in FIG. 1, the BPF receives the signal V(t) and generates the signal R(t) which is ready to undergo amplification and transmission over a communications channel. The amplification is assumed linear, otherwise, the conversion filter will have to be predistored in order to compensate for distortion inflicted on R(t) due to nonlinear amplification. The BPF has a nonzero flat response over the frequency range [f_(max) -f_(co) f_(max) +f_(co) ] where f_(max) corresponds to the positive frequency and a zero response elsewhere the power spectral density of S(t) reaches its maximum and f_(co) is the cutoff frequency of the BPF.

EXPLANATION OF THE INVENTION

The following explanation is in relation to the digital signaling system in FIG. 1. By increasing the spread L of the impulse response of the conversion filter, the Power Spectral Density (PSD) of the signal S'(t), which is defined as: ##EQU3## is reduced (assuming that β consists of equiprobable symbols). In other words, by increasing L, the bandwidth efficiency is increased compared to a conventional linear digital signaling system which uses a shaping filter satisfying Nyquist criterion. The power spectral density of S'(t) corresponds to the complex autocorrelation.

By increasing the spread L of the conversion filter however, memory is added into S'(t). In other words, the amount of Intersymbol Interference (ISI) that exists in S'(t) is directly proportional to L. Such an ISI can in turn decrease the value of d_(min) ² /E_(b) below the value 4.0 which corresponds to the maximum value of d_(min) ² /E_(b) a conventional linear digital signaling system can reach. Nontheless, ranges of the modulation index h of the conversion filter exist where d_(min) ² /E_(b) is increased by varying h.

Finally, S(t) is not necessarily bandlimited. To bandlimit S(t), it must be filtered using a bandpass filter as shown in FIG. 1. In this case, the bandpass filter has to be centered in the frequency domain around f_(max) and frequency f_(co) has to be chosen large enough to pass essentially about 90% or more of the average power in the signal S(t).

The present invention describes a transmission process and apparatus which is, of course, only one end of the transmitter/receiver pair. Receivers that would have practical utility with the present invention are well known in the art, as described, for example, in J. G. Proakis "Digital Communications," McGraw-Hill, New York, 1983, and would include the following parts: a coherent demodulator, followed by a filter matched to "g(t)", possibly an equalizer to remove part of the ISI introduced by the BPF and the channel, and finally a Viterbi Algorithm (VA). However, suboptimal receives can also be used at the expenses of a poorer performance of the receiver. Examples of suboptimal receivers are (as described in the last mentioned reference) non-coherent receivers (such as differential detectors or frequency discriminators to be used when coherent detection is not possible) and symbol-by-symbol receivers (such as Linear Mean Square Equalizers or Decision Feedback Equalizers to be used when low complexity receivers are desirable).

A preferred receiver would consist of a coherent demodulator, followed by a filter matched to "g(t)" and a Viterbi Algorithm (VA). The VA would preferably have the following complexity: "M^(L) " states/2-Dimensions if β consists of real symbols and "M^(2L) " states/2-Dimensions if β consists of complex symbols. A VA with a smaller number of states can also be used at the expenses of a poorer performance of the receiver.

See attached software for a display of the preferred Viterbi Algorithm. The software consists of a computer simulation of the coherent receiver consisting of a matched filter and a VA (VitAlg.f, vitmin.f, vitmin Q.f, vitdecod.f, vitdecodQ.f) according to the above preferred specifications of the bandpass filter (channel WMF.f) over an Additive White Gaussian Noise channel.

With a given encoder and modulator, the various parameters of the conversion filter and of the bandpass filter may be chosen as follows:

1. The spread L of the conversion filter is chosen according to the desired BW efficiency with f_(co) chosen such that about 90% or more of the average power in the signal S(t) output from the conversion filter is allowed through the BPF.

2. The modulation index h is chosen according to L such that d_(min) ² /E_(b) reaches a local maximum. One has to realize that by fixing the bandwidth efficiency, L is fixed and consequently, the complexity of the preferred VA is also fixed.

EXAMPLES OF THE PREFERRED PARAMETERS

The following examples demonstrate the preferred values for the parameters L, σ, h and f_(co) according to the steps mentioned in the previous section, together with the corresponding d_(min) ² /E_(b) and BW efficiency. When a Viterbi Algorithm (VA) is used in the receiver, its preferred number of states/2 Dimensions is also shown in the following examples.

EXAMPLE I

The encoder consists of an M-ary mapper where β is a sequence of real symbols that take one of the M values: {±1, ±3, . . . , ±(M-1)}, followed by a Gray encoder. Table I summarizes the preferred values of σ and f_(co) for L=2, . . . , 9, h=1/4 and M=2 and 4, together with the corresponding d_(min) ² /E_(b), BW efficiency and the number of states required in a VA.

                  TABLE I                                                          ______________________________________                                                                  d.sub.min.sup.2                                                                      BW efficiency                                                                            # of VA                               M    L      σT                                                                               f.sub.co T                                                                          /E.sub.b                                                                             bits/s/Hz states/2D                             ______________________________________                                         2    2      0.200   0.500                                                                               4.00  1.0        2                                    2    3      0.120   0.277                                                                               2.70  1.8        4                                    2    4      0.090   0.208                                                                               1.82  2.4        8                                    2    5      0.070   0.166                                                                               1.22  3.0       16                                    2    6      0.060   0.139                                                                               0.82  3.6       32                                    2    7      0.050   0.116                                                                               0.56  4.3       64                                    2    8      0.044   0.102                                                                               0.38  4.9       128                                   2    9      0.039   0.091                                                                               0.25  5.5       256                                   4    2      0.200   0.500                                                                               1.60  2.0        4                                    4    3      0.120   0.277                                                                               1.08  3.6       16                                    4    4      0.090   0.208                                                                               0.73  4.8       64                                    4    5      0.070   0.166                                                                               0.25  6.0       256                                   ______________________________________                                    

One should notice that for Binary Phase Shift Keying, d_(min) ² /E_(b) is equal to 4.0 which represents the maximum value d_(min) ² /E_(b) can reach for a linear modulation scheme.

EXAMPLE II

Once again, the encoder consists of an M-ary mapper and a Gray encoder. In this case however, β consists of complex symbols (i.e. the imaginary part of β_(i) takes one of the M values: {±1, ±3, . . . , ±(M-1)}). Table II summarizes the preferred values of σ and f_(co) for L=2, . . . , 5, h=1/4 and M=2 and 4, together with the corresponding d_(min) ² /2E_(b), the BW efficiency and the number of states required in a VA.

                  TABLE II                                                         ______________________________________                                                                  d.sub.min.sup.2                                                                      BW efficiency                                                                            # of VA                               M    L      σT                                                                               f.sub.co T                                                                          /E.sub.b                                                                             bits/s/Hz states/2D                             ______________________________________                                         2    2      0.200   0.500                                                                               4.00  2.0        4                                    2    3      0.120   0.277                                                                               2.70  3.6       16                                    2    4      0.090   0.208                                                                               0.82  4.8       64                                    2    5      0.070   0.166                                                                               0.25  6.0       256                                   4    2      0.200   0.500                                                                               1.60  4.0       16                                    4    3      0.120   0.277                                                                               1.00  7.2       256                                   ______________________________________                                    

It will be understood that while preferred embodiments of the invention have been described, immaterial modifications may be made to the invention without departing from its substance and these are intended to be covered by the scope of the invention.

Further summary of the invention are found in the claims that follow. ##SPC1## 

I claim:
 1. Apparatus for transmitting a digital signal over a communications channel, the apparatus comprising:at least one encoder for converting digital signals into a sequence of symbols having a signalling interval T; a conversion filter connected to the encoder, the conversion filter having an impulse response with a spread L having a value between 2 and 9 for filtering the sequence of symbols to generate a signal S, in which S is dependent on the signalling interval T; a modulator connected to the conversion filter for modulating the signal S to generate a signal V; a bandpass filter connected to the modulator, the bandpass filter having a cut-off frequency f_(co) for filtering the signal V to generate a signal R; in which f_(co) and T are chosen so that the product of f_(co) and T is less than or equal to 1 and greater than or equal to 0.182.
 2. The apparatus of claim 1 in which the impulse response of the conversion filter is defined as a Gaussian function of time, and has the form:

    g(t)=(4πσ.sup.2).sup.1/4 e.sup.-2(πσ) 2 e.sup.j2πht/T

in which σ is inversely proportional to L.
 3. The apparatus of claim 2 in which the encoder is an M-ary mapper followed by a Gray encoder, in which the symbols take only real values, in which the conversion filter has a modulation index h, h being equal to 1/4, and M, L, the product of T and σ, and the product of T and f_(co) having values as shown in the rows of the following table:

    ______________________________________                                         M       L              σT                                                                              f.sub.co T                                       ______________________________________                                         2       2              0.200  0.500                                            2       3              0.120  0.277                                            2       4              0.090  0.208                                            2       5              0.070  0.166                                            2       6              0.060  0.139                                            2       7              0.050  0.116                                            2       8              0.044  0.102                                            2       9              0.039  0.091                                            4       2              0.200  0.500                                            4       3              0.120  0.277                                            4       4              0.090  0.208                                            4       5              0.070  0.166                                            ______________________________________                                    


4. The apparatus of claim 2 in which the encoder is an M-ary mapper followed by a Gray encoder, in which the symbols take complex values, in which the conversion filter has a modulation index h, h being equal to 1/4 and M, L, the product of T and σ, and the product of T and f_(co) having values as shown in the rows of the following table:

    ______________________________________                                         M       L              σT                                                                              f.sub.co T                                       ______________________________________                                         2       2              0.200  0.500                                            2       3              0.120  0.277                                            2       4              0.090  0.208                                            2       5              0.070  0.166                                            4       2              0.200  0.500                                            4       3              0.120  0.277                                            ______________________________________                                    


5. The apparatus of claim 1 in which the bandpass filter has a nonzero flat response over the frequency range between f_(max) -f_(co) and f_(max) +f_(co).
 6. A process for transmitting a digital signal over a communications channel, the process comprising the steps of:encoding the digital signals to generate a sequence of symbols having a signalling interval T; filtering the sequence of symbols with a conversion filter to generate a signal S, the conversion filter having an impulse response having a spread L having a value between 1 and 9 and a modulation index h; generating a real signal V from the signal S using a modulator; filtering the signal V with a bandpass filter to generate a signal R, the bandpass filter having a cut-off frequency f_(co) ; in which f_(co) and T are chosen so that the product of f_(co) and T is less than or equal to 1 and greater than or equal to 0.182.
 7. The process of claim 6, in which the impulse response of the conversion filter is defined as a Gaussian function of time.
 8. The process of claim 6 in which the bandpass filter has a nonzero flat response over the frequency range between f_(max) -f_(co) and f_(max) +f_(co).
 9. The process of claim 6 in which the encoder is an M-ary mapper followed by a Gray encoder, in which the symbols take only real values, in which the impulse response of the conversion filter is defined as a Gaussian function of time having the form g(t)=(4πσ²)^(1/4) e⁻²(πσ)2 e^(j2)πht/T where σ is inversely proportional to L, in which the conversion filter has a modulation index h equal to 1/4 and M, L, the product of T and σ, and the product of T and f_(co) having values as shown in the rows of the following table:

    ______________________________________                                         M       L              σT                                                                              f.sub.co T                                       ______________________________________                                         2       2              0.200  0.500                                            2       3              0.120  0.277                                            2       4              0.090  0.208                                            2       5              0.070  0.166                                            2       6              0.060  0.139                                            2       7              0.050  0.116                                            2       8              0.044  0.102                                            2       9              0.039  0.091                                            4       2              0.200  0.500                                            4       3              0.120  0.277                                            4       4              0.090  0.208                                            4       5              0.070  0.166                                            ______________________________________                                    


10. The process of claim 6 in which the encoder is an M-ary mapper followed by a Gray encoder, in which the symbols take complex values, in which the impulse response of the conversion filter is defined as a Gaussian function of time having the form g(t)=(4πσ²)^(1/4) e⁻²(πσ)2 e^(j2)πht/T where σ is inversely proportional to L, in which the conversion filter has a modulation index h equal to 1/4 and M, L, the product of T and σ, and the product of T and f_(co) having values as shown in the rows of the following table:

    ______________________________________                                         M       L              σT                                                                              f.sub.co T                                       ______________________________________                                         2       2              0.200  0.500                                            2       3              0.120  0.277                                            2       4              0.090  0.208                                            2       5              0.070  0.166                                            4       2              0.200  0.500                                            4       3              0.120  0.277                                            ______________________________________                                     